Electronic Communications in Probability

Short proofs in extrema of spectrally one sided Lévy processes

Loïc Chaumont and Jacek Małecki

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Abstract

We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall’s identity for spectrally positive Lévy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall’s identity, Theorem 2.4 in [20] which gives an expression for the law of the supremum of spectrally positive Lévy processes. An analogous formula is obtained for the supremum of spectrally negative Lévy processes.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 55, 12 pp.

Dates
Received: 17 April 2018
Accepted: 10 August 2018
First available in Project Euclid: 1 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1535767266

Digital Object Identifier
doi:10.1214/18-ECP163

Mathematical Reviews number (MathSciNet)
MR3852269

Zentralblatt MATH identifier
1398.60068

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60G09: Exchangeability

Keywords
cyclically interchangeable process spectrally one sided Lévy process Ballot theorem Kendall’s identity past supremum bridge

Rights
Creative Commons Attribution 4.0 International License.

Citation

Chaumont, Loïc; Małecki, Jacek. Short proofs in extrema of spectrally one sided Lévy processes. Electron. Commun. Probab. 23 (2018), paper no. 55, 12 pp. doi:10.1214/18-ECP163. https://projecteuclid.org/euclid.ecp/1535767266


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